Intersection of Two Lines

Let $\ell_1$ and $\ell_2$ now be two lines with parameters $\mathbf{l}_1$ and $\mathbf{l}_2$ intersecting at the point $\mathbf {x}$ expressed in homogeneous coordinates. To obtain the intersection point, it is necessary to solve a homogeneous system of the form

$$\left\{ \begin{array}{rl} \mathbf{l}_1^{\top} \mathbf{x} &... ... \mathbf{l}_2^{\top} \mathbf{x} & = 0 \\ \end{array} \right.$$ (1.75)

The system, of type $\mathbf{A}\mathbf{x}=0$, can also be extended to the case of n intersecting lines, with $n > 2$, yielding an overdetermined system that can be solved using SVD or QR decomposition. The solution of the overdetermined system, affected by noise, represents the point that minimizes the algebraic residual of equation (1.75).

In the case of only two lines, system (1.75) directly provides the solution. The intersection of two lines $\mathbf{l}_1$ and $\mathbf{l}_2$, written in implicit form (1.59), is the point $\mathbf{x} = \mathbf{l}_1 \times \mathbf{l}_2$ expressed in homogeneous coordinates, where $\times$ is the cross product.

It should be noted that, since homogeneous coordinates can represent points at infinity, this formalism also admits the case in which the two lines are parallel.